On Pinned Falconer Distance Problem for Cartesian Product Sets: the Parabolic Method
Ji Li, Chong-Wei Liang, Chun-Yen Shen
TL;DR
The paper advances the pinned Falconer distance problem for Cartesian product sets by introducing a parabolic-structure approach that leverages a curvature-conditioned distance function. By applying a Parabolic distance framework with Phong–Stein rotation curvature and Sogge cinematic curvature to carefully constructed product-sets E and F, the authors translate lower bounds on the parabolic pinned distance to the classical pinned distance for A^d, obtaining improved dimension- or measure-thresholds. They perform a three-case analysis based on the dimension of (A∩B)^2, deriving explicit conditions under which the pinned distance set has Hausdorff dimension at least β, positive measure, or contains an interval, and they specialize these results to A^d and to general Cartesian products. The results extend prior work by Iosevich–Liu and Erdogan, providing stronger thresholds in certain regimes (notably when s_{d-1}+s_d>5/4) and suggesting the parabolic-method framework can enhance other distance-variant problems on product sets.
Abstract
The Falconer distance problem for Cartesian product sets was introduced and studied by Iosevich and Liu (\cite{MR3525385}). In this paper, by implementing a new observation on Cartesian product sets associated with a particular parabolic structure, we study the pinned version of Falconer distance problem for Cartesian product sets, and improve the threshold for the Falconer distance set in \cite{MR3525385} in certain case.